Phase transitions in physical systems have been an active area of scientific investigation for many years. Solid/liquid/vapor transitions such as those that occur between ice, water, and steam are undoubtedly familiar examples. Another common example is sugar crystal formation in honey that has been left to stand for a long time. Phase transitions, however, occur in many other types of systems—for example, the transition from ferromagnetic to paramagnetic behavior for ferromagnets at the Curie temperature, or the transition from normal conductivity to superconductivity in certain metals and ceramic oxides at a critical temperature, or the conductor/insulator percolation threshold for electrical networks.
Phase transitions are classified as being either first-order or continuous (or, more infrequently, second-order) phase transitions. First-order transitions such as those between ice, water, and steam, involve the evolution or absorption of heat at the transition point. Continuous transitions, by contrast, are not accompanied by heat transfer. In addition, continuous phase transitions are accompanied by the growth of fluctuations on ever-longer length scales. Transitions are further organized into groupings called universality classes. For systems from the same universality class, renormalizing raw experimental data to the proper critical conditions has the remarkable effect of “collapsing” data onto what is essentially a master curve. FIG. 1 shows two examples of this phenomenon: FIG. 1A demonstrates that pressure/volume/temperature data for several different fluids fall on top of one another when the data are properly renormalized to the critical point; FIG. 1B illustrates that the same type of collapse occurs for ferromagnetic materials near the Curie temperature. See J. M. Yeomans, The Statistical Mechanics of Phase Transitions, (1992) p. 28 for FIG. 1A, and p. 119 for FIG. 1B.
Modeling of phase transitions is important for designing distillation columns using random or structured packings. In normal operation, packed columns are operated countercurrently with the vapor as the continuous phase and the liquid as the dispersed phase. These columns can reach points of hydraulic inoperability generally referred to as “flood points.” Flooding is typically associated with large fluctuations in the pressure drop, an abrupt increase in the liquid holdup and the pressure drop, and excessive liquid entrainment. Investigators have noted that flooding seems to be associated with a transition from vapor-continuous to liquid-continuous operation. See Eckert, J. S. New Look at Distillation-4 Tower Packings . . . Comparative Performance, Chem. Eng. Prog. 59 pp. 76-82 (May 1963). FIG. 2 shows the buildup of liquid on the top of a packed column near the flood point.
The pressure drop of a vapor flowing countercurrently upward relative to the liquid flow is illustrated in FIG. 3. For all liquid rates, a zone is reached where the pressure drop breaks upward. Column instability sets in at the points B, B′ and reaches a maximum at C, C′, where the increase in pressure drop is quite large with only a small increase in the vapor flow rate. Points C and C′ are referred to as flood points, and it is here that liquid accumulation on the top of a packed bed usually becomes visually apparent, as illustrated in FIG. 2.
The flooding phenomenon in packed columns is extremely complex. It is possible to operate a column away from flooding in either vapor-continuous or liquid-continuous mode. The crossover from normal vapor-continuous operation to something more akin to operation in a liquid-continuous mode is signaled by the onset of a change in slope of the pressure drop versus vapor velocity from the slope of the curves away from flooding. In FIG. 4, the dashed lines have been drawn with the same slope as the dry (zero liquid flow) pressure drop curve. S. M. Walas, Chemical Process Equipment: Selection and Design, Butterworth-Heinemann (2002). At a liquid loading of 20,000 lbs/hr-ft2, the initial slope (between 0.1 and 0.2 “H2O/ft (inches of water per foot of column packing)) is virtually identical to that of the dry pressure drop curve. At 30,000 lbs/hr-ft2, however, the initial slope has become noticeably different, signaling the transition to operation in a liquid-continuous mode.
For maximum product output from a distillation column, it is desirable to operate in vapor-continuous mode (with the gas as the vapor-continuous phase and the liquid as the dispersed phase) at the highest gas and liquid flow rates achievable without flooding the column. To assist in column design, packing manufacturers provide air/water pressure drop data at different liquid flow rates, such as the example shown in FIG. 4. There is a need, however, for a method of generating a unified mathematical expression of pressure drop in a packed column at any liquid flow rate.